Extrema: maximum and minimum

Let's see now the definitions of relative and absolute extrema by means of an example:

• A function f has a relative or local minimum (respectively relative or local maximum) in $x = a$, if there is a neighbourhood $E_r(a)$ of the point, such that for every $x$ belonging to $E_r(a)$, we have $$f(x) \leq f(a) (\mbox{ respectively }f(x) \geq f(a))$$

• A function $f$ has an absolute minimum (respectively absolute maximum) in $x = a$, if for any $x$ of the domain of $f$ we have $$f(x) \leq f(a) (\mbox{ respectively }f(x) \geq f(a))$$

Consider the following function:

We observe that it has:

• Relative maximum in $x =-1$ relative minimum in $x = 1$
• The function is odd and it is not bounded.